Constant Growth Model: Stock Sells For 40 Next Dividend ✓ Solved

15 Constant Growth Model A Stock Sells For 40 The Next Dividend W

Analyze various scenarios involving the constant growth dividend discount model (DDM), including calculating stock prices, expected returns, and growth rates based on given financial data. The problems involve understanding how dividends grow over time, how to determine required rates of return, and interpreting market data such as risk premiums and historical return statistics.

The assignment includes calculations related to stock valuation, expected return estimates, impact of changing discount rates, and the relationship between dividend growth and stock prices. It also explores the implications of historical market data and risk premiums on investment decision-making. These analyses aim to deepen understanding of the constant growth model and its applications within stock valuation and investment strategies.

Paper For Above Instructions

The constant growth dividend discount model (DDM) serves as a fundamental valuation method in finance, allowing investors to estimate the intrinsic value of a stock based on its expected future dividends that grow at a constant rate. This paper covers scenarios where the model is applied to various stocks with given dividend payments, growth rates, and market conditions, illustrating the practical use of the model and the interplay of different financial variables.

Determining the Required Rate of Return

Consider the scenario where a stock sells for $40, and the next expected dividend is $4. The key is to estimate the discount rate, which reflects the total expected return for an investor (including growth and income). Given the dividend growth rate, reinvestment rate, and the company's reinvestment policy, the required rate of return can be derived using the DDM formula:

P = D1 / (r - g)

Where:

- P = current stock price ($40),

- D1 = next dividend ($4),

- r = required rate of return,

- g = growth rate of dividends.

Calculating the Growth Rate for the Stock

In cases where the dividend growth rate isn’t explicitly given, it can be inferred from reinvestment and earnings growth. A company reinvesting 40% of earnings and earning a 15% return on reinvestments suggests a growth rate (g) calculated as:

g = Reinvestment rate × Return on reinvested earnings = 0.40 × 0.15 = 0.06 or 6%.

Applying the Dividend Discount Model

Using the DDM formula and the outlined assumptions, the required rate of return (r) for the stock in this case can be computed as follows:

Rearranging the formula:

r = (D1 / P) + g = ($4 / $40) + 0.06 = 0.10 + 0.06 = 0.16 or 16%.

This indicates an expected total return of 16%. The small difference between the actual dividend yield (10%) and the total expected return indicates a dividend growth component (6%) consistent with the company's reinvestment policy.

Valuation and Sensitivity to Discount Rates

Suppose we analyze a stock like Gentleman Gym paying a $3 dividend with an annual growth of 5%. With a discount rate of 15%, the stock’s value (P) can be calculated as:

P = D1 / (r - g) = ($3 × 1.05) / (0.15 - 0.05) = $3.15 / 0.10 = $31.50.

If the discount rate drops to 12%, the valuation increases to:

P = $3.15 / (0.12 - 0.05) = $3.15 / 0.07 ≈ $45.

This demonstrates how a lower discount rate leads to a higher stock valuation, reflecting increased investor confidence or lower opportunity cost of capital.

Estimating the Expected Return and Growth Rate

For Eastern Electric with a recent dividend of $1.64 and a stock price of $27, assuming a 3% dividend growth, the expected return (r) can be computed via:

r = (D1 / P) + g = ($1.64 × 1.03) / $27 + 0.03 ≈ ($1.69) / $27 + 0.03 ≈ 0.0626 + 0.03 ≈ 0.0926 or 9.26%.

Similarly, if investors require a 10% return, the implied growth rate is:

g = r - (D1 / P) = 0.10 - 0.0607 ≈ 0.0393 or 3.93%, aligning with the sustainable growth assumptions.

If the company has a plowback ratio of 0.4 and its return on new investments is needed to sustain a 5% growth rate, the return on investments (ROE) is:

g = b × ROE → 0.05 = 0.4 × ROE → ROE = 0.125 or 12.5%.

Valuation of Stock with Given Dividends and Growth Expectations

For the Non-Stick Gum Factory, expecting a dividend of $2 next year with a 6% growth rate and a required return of 12%, the valuation is:

P = D1 / (r - g) = $2 / (0.12 - 0.06) = $2 / 0.06 = approximately $33.33.

This valuation reflects the present value considering future growth and required return parameters.

Historical Market Data and Risk Premiums

Analyzing average returns on U.S. stocks from 1900 to 2017 shows a long-term average return of approximately 10-12%, with a risk premium of 4-6% over risk-free rates, typically around 2% during these periods. During the 1977–1981 period, when bonds yielded negative returns, this indicates a period of high inflation and interest rate volatility, causing temporary deviations from expected risk premiums.

Negative bond returns suggest investors priced in heightened inflation expectations or economic instability, but such periods are generally transient. Over the long term, the default assumption is a positive maturity premium as investors demand compensation for bearing interest rate risk.

Impact of Market Volatility

If next year's stock market return drops to -20%, it indicates increased market risk or economic downturns, potentially causing a reevaluation of the "normal" risk premium, which might decrease temporarily due to heightened risk aversion. Such negative swings also underline the importance of diversification and prudent risk assessment in investment strategies.

Valuation Using the Capital Asset Pricing Model (CAPM)

Assuming a stock with the same market risk as the S&P 500 is expected to sell at $50, with a dividend of $2, and a risk-free rate of 2%, the appropriate discount rate can be estimated via the CAPM:

r = Rf + β (Rm - Rf)

Where Rf = 2%, Rm - Rf (market risk premium) ≈ 8-10%, and β = 1 for the S&P 500. Plugging in these numbers:

r ≈ 0.02 + 1 × 0.08 = 0.10 or 10%

Thus, the fair price today can be calculated as:

P0 = (Dividend / (r - g)) = $2 / (0.10 - 0.06) = $2 / 0.04 = $50.

This aligns with the market expectation and the given future sale price, confirming the valuation consistency.

Conclusion

The application of the constant growth dividend discount model in these scenarios highlights the intricacies of stock valuation, the importance of accurately estimating growth rates, required returns, and market premiums. Understanding how market conditions influence these variables is crucial for both investors and analysts in making informed investment decisions. Recognizing the effects of historical data, risk premiums, and macroeconomic factors allows for more nuanced and robust valuation models, ultimately leading to better portfolio management and investment outcomes.

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